Total expenses of living in a boarding house are partly fixed and partly vary linearly with the number of people living there. The average monthly rent per person is Rs. 1650 when the total number of people living there is 50, and it is Rs. 1320 when 100 people live there. If the monthly rent of each individual includes a 10% markup over the cost, how much does it cost the boarding house to host each person when the people living there are 125?

Let's break down the problem and solve it step by step.

### Step 1: Formulating the equation for total expenses
The total expenses \( T \) of living in the boarding house consist of two components:
- A fixed expense component, \( F \), which does not depend on the number of people.
- A variable expense component that depends linearly on the number of people. Let the variable cost per person be \( v \).

So, the total expense \( T \) for \( n \) people is given by:
\[
T = F + n \cdot v
\]

### Step 2: Average rent per person
The average monthly rent per person \( R \) is:
\[
R = \frac{T}{n} = \frac{F + n \cdot v}{n} = \frac{F}{n} + v
\]

### Step 3: Using the given data
We are given two pieces of information about the rent per person:
1. When there are 50 people living there, the average rent per person is Rs. 1650.
   \[
   R_{50} = \frac{F}{50} + v = 1650
   \]
2. When there are 100 people living there, the average rent per person is Rs. 1320.
   \[
   R_{100} = \frac{F}{100} + v = 1320
   \]

### Step 4: Solving for \( F \) and \( v \)
We now have two equations:
1. \(\frac{F}{50} + v = 1650\)
2. \(\frac{F}{100} + v = 1320\)

We can solve this system of equations to find \( F \) (the fixed cost) and \( v \) (the variable cost per person).

Subtract the second equation from the first:
\[
\left( \frac{F}{50} + v \right) - \left( \frac{F}{100} + v \right) = 1650 - 1320
\]
This simplifies to:
\[
\frac{F}{50} - \frac{F}{100} = 330
\]
\[
\frac{F}{100} = 330
\]
\[
F = 330 \times 100 = 33,000
\]

Now substitute \( F = 33,000 \) into one of the original equations to solve for \( v \). Using the first equation:
\[
\frac{33,000}{50} + v = 1650
\]
\[
660 + v = 1650
\]
\[
v = 1650 - 660 = 990
\]

So, the fixed cost \( F = 33,000 \) and the variable cost per person \( v = 990 \).

### Step 5: Calculating the cost for 125 people
The total cost for 125 people is:
\[
T = F + 125 \cdot v = 33,000 + 125 \cdot 990
\]
\[
T = 33,000 + 123,750 = 156,750
\]

The total cost of hosting 125 people is Rs. 156,750.

### Step 6: Removing the markup
We are told that the monthly rent includes a 10% markup over the cost. Therefore, the cost per person (without markup) is:
\[
\text{Cost per person} = \frac{\text{Rent per person}}{1.10}
\]

The rent per person is:
\[
R_{125} = \frac{156,750}{125} = 1254
\]

Now, removing the 10% markup:
\[
\text{Cost per person} = \frac{1254}{1.10} = 1140
\]

### Final Answer:
The cost to the boarding house to host each person when there are 125 people is Rs. 1140.

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Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How would the cost per person change if there were 150 people instead of 125?
2. How does the fixed component affect the rent if the number of people changes significantly?
3. Can you derive a general formula to calculate the cost per person for any number of people?
4. What would happen to the variable cost per person if the expenses started increasing exponentially instead of linearly?
5. If the markup were 15% instead of 10%, how would that affect the rent calculation?

**Tip**: When solving cost-related problems with both fixed and variable components, breaking the problem into smaller equations can make the solution clearer.

Learn how to calculate the cost of hosting each person in a boarding house when total expenses are partly fixed and partly vary linearly with the number of residents. This problem involves solving linear equations to determine fixed and variable costs, and applying a 10% markup removal. Suitable for Grades 10-12.

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